Existence and uniqueness of a finite energy solution for the mixed value problem of porous thermoelastic bodies

We consider the mixed problem with boundary and initial data in thermoelasticity of porous bodies with dipolar structure. By generalizing some known results developed by Dafermos in a more simple case of the classical theory of elasticity, we prove new theorems in which we address the issues regarding the uniqueness and existence of a solution with finite energy of the respective problem after we define this type of solution.

Existence and decay of finite energy solutions for semilinear dissipative wave equations in time-dependent domains

We consider the initial-boundary value problem for semilinear dissipative wave equations in noncylindrical domain \(\bigcup_{0\leq t \lt\infty} \Omega(t)\times\{t\} \subset \mathbb{R}^N\times \mathbb{R}\). We are interested in finite energy solution. We derive an exponential decay of the energy in the case \(\Omega(t)\) is bounded in \(\mathbb{R}^N\) and the estimate \[\int\limits_0^{\infty} E(t)dt \leq C(E(0),\|u(0)\|)< \infty\] in the case \(\Omega(t)\) is unbounded. Existence and uniqueness of finite energy solution are also proved.

Solutions with Prescribed Local Blow-up Surface for the Nonlinear Wave Equation

We prove that any sufficiently differentiable space-like hypersurface of {{\mathbb{R}}^{1+N}} coincides locally around any of its points with the blow-up surface of a finite-energy solution of the focusing nonlinear wave equation {\partial_{tt}u-\Delta u=|u|^{p-1}u} on {{\mathbb{R}}\times{\mathbb{R}}^{N}}, for any {1\leq N\leq 4} and {1<p\leq\frac{N+2}{N-2}}. We follow the strategy developed in our previous work (2018) on the construction of solutions of the nonlinear wave equation blowing up at any prescribed compact set. Here to prove blow-up on a local space-like hypersurface, we first apply a change of variable to reduce the problem to blowup on a small ball at {t=0} for a transformed equation. The construction of an appropriate approximate solution is then combined with an energy method for the existence of a solution of the transformed problem that blows up at {t=0}. To obtain a finite-energy solution of the original problem from trace arguments, we need to work with {H^{2}\times H^{1}} solutions for the transformed problem.

Regularizing effect for a system of Schrödinger–Maxwell equations

We prove some existence results for the following Schrödinger–Maxwell system of elliptic equations:\left\{\begin{aligned} &\displaystyle{-}\div(M(x)\nabla u)+A\varphi|u|^{r-2}u=% f,&&\displaystyle u\in W_{0}^{1,2}(\Omega),\\ &\displaystyle{-}\div(M(x)\nabla\varphi)=|u|^{r},&&\displaystyle\varphi\in W_{% 0}^{1,2}(\Omega).\end{aligned}\right.In particular, we prove the existence of a finite energy solution {(u,\varphi)} if {r>2^{*}} and f does not belong to the “dual space” {L^{\frac{2N}{N+2}}(\Omega)}.

The Cahn–Hilliard–Oono equation with singular potential

We consider the so-called Cahn–Hilliard–Oono equation with singular (e.g. logarithmic) potential in a bounded domain of [Formula: see text], [Formula: see text]. The equation is subject to an initial condition and Neumann homogeneous boundary conditions for the order parameter as well as for the chemical potential. However, contrary to the Cahn–Hilliard equation, the total mass might not be conserved. The existence of a global finite energy solution to such a problem was proven by Miranville and Temam. We first establish some regularization properties in finite time of the (unique) solution. Then, in dimension two, we prove the so-called strict separation property, namely, we show that any finite energy solution stays away from pure phases, uniformly with respect to the initial energy and the total mass. Taking advantage of these results, we study the long-time behavior of solutions. More precisely, we establish the existence of the global attractor in both two and three dimensions. Due to the strict separation property in dimension two, we also prove the existence of exponential attractors and we show that a finite energy solution always converges to a single equilibrium even though the mass is not conserved.

SOME EXACT BLOWUP SOLUTIONS TO MULTIDIMENSIONAL SCHRÖDINGER MAP EQUATION ON HYPERBOLIC SPACE AND CONE

Exact solutions for the multidimensional Schrödinger map equation (SM for short) on hyperbolic 2-space [Formula: see text] cone are obtained. Consequently, we show the non-traveling wave solution on [Formula: see text] is a finite energy solution on the finite spacial domain. The question of whether a solution of SM can develop a finite time singularity on [Formula: see text] with smooth initial data is not clear. Our result show that blowup can really happen on this initial data. In addition, some exact global smooth solutions are constructed.

Null controllability of a thermoelastic plate

Thermoelastic plate model with a control term in the thermal equation is considered. The main result in this paper is that with thermal control, locally distributed within the interior and square integrable in time and space, any finite energy solution can be driven to zero at the control timeT.